Clear Notion
Login

Practice

Beyond Advanced · Statistics and Probability

Your Score

0 / 280

Correct

0 / 70

Attempted

0 / 70

1
Seven observations are recorded. Each observation is an integer from 00 to 1010 (inclusive), repetitions are allowed, and the mean of the seven observations is 247\frac{24}{7}. Find the largest possible value of the variance of the seven observations (variance =1n∑(xi−xˉ)2=\frac{1}{n}\sum (x_i-\bar x)^2). Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 19.1
2
Ten real numbers x1,x2,…,x10x_1, x_2, \dots, x_{10} have variance 99. From them form the 4545 numbers xi+xjx_i+x_j (1≤i<j≤10)(1\le i<j\le 10), one for each unordered pair. What is the variance of these 4545 numbers?
Not attempted
A
1616
B
1818
C
725\frac{72}{5}
D
2020
3
For a set of n≥2n\ge 2 real observations, let xˉ\bar x be the mean, MM the median, σ\sigma the standard deviation (with divisor nn) and RR the range (largest minus smallest). Which of the following statements hold for EVERY such data set?
Not attempted
A
If the data have a unique mode ZZ, then ∣xˉ−Z∣≤σ|\bar x-Z|\le\sigma
B
σ≤R2\sigma\le\frac{R}{2}
C
σ≥R2n\sigma\ge\frac{R}{\sqrt{2n}}
D
∣xˉ−M∣≤σ|\bar x-M|\le\sigma
4
In the data set shown in the bar chart, the value kk occurs exactly kk times, for each k=1,2,…,20k=1,2,\dots,20 (so there are 210210 observations). Find the mean deviation of this data set about its median.
Not attempted
kfrequency12345678910111213141516171819205101520
Your answer: —
Correct answer: 4
5
Five distinct positive integers have mean 1010 and median 1212. Find the smallest possible value of their variance (divisor 55).
Not attempted
A
745\frac{74}{5}
B
22
C
825\frac{82}{5}
D
1414
6
The histogram shows the marks of 7070 students in the classes 00-1010, 1010-2020, 2020-3030, 3030-4040, 4040-5050 with frequencies 8, f1, 25, f2, 78,\ f_1,\ 25,\ f_2,\ 7 respectively (f1,f2f_1, f_2 are unknown positive integers, the dashed bars are not to scale). The class 2020-3030 is the modal class, and the mode and the median of the distribution (by the usual grouped-data formulae) are equal. Find the mean of the distribution (using class marks).
Not attempted
marksfrequency010203040508257f₁f₂
A
2424
B
1697\frac{169}{7}
C
1777\frac{177}{7}
D
2525
7
The figure shows the 'less than' ogive (rising) and the 'more than' ogive (falling) of a grouped frequency distribution with class intervals 00-2020, 2020-4040, 4040-6060, 6060-8080, 8080-100100. The plotted cumulative frequencies are, at x=0,20,40,60,80,100x=0,20,40,60,80,100: less than: 0,6,20,44,64,800,6,20,44,64,80; more than: 80,74,60,36,16,080,74,60,36,16,0. Which of the following statements are correct?
Not attempted
xcum. freq.O020406080100061620364460647480
A
The two ogives meet at a point whose ordinate is 4040
B
The mode of the distribution (grouped-data formula) is 3807\frac{380}{7}
C
The abscissa of the point where the ogives meet is 1703\frac{170}{3}
D
The mean of the distribution (using class marks) is larger than its median
8
Find the minimum value of f(t)=∑k=110∣kt−1∣f(t)=\sum_{k=1}^{10}|kt-1| over all real tt. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 3.86
9
Five real numbers have mean 66. The sum of the squares of all 1010 pairwise differences, ∑i<j(xi−xj)2\sum_{i<j}(x_i-x_j)^2, equals 200200. Two more numbers, each equal to 66, are now added to the data. What is the standard deviation of the seven numbers?
Not attempted
A
207\sqrt{\frac{20}{7}}
B
222\sqrt2
C
407\sqrt{\frac{40}{7}}
D
2007\sqrt{\frac{200}{7}}
10
List-IList-IIP) Variance of the data in which the value k occurs (8k) times, k=0,1,…,81) 2Q) Mean deviation about the mean of the data in P2) 3532R) Variance of the data in which the value k occurs k times, k=1,2,…,103) 6S) Variance of the 15 numbers a+b, over all pairs a<b from {1,2,…,6}4) 1435) 3512\begin{array}{ll}\textbf{List-I} & \textbf{List-II} \\ \text{P) Variance of the data in which the value } k \text{ occurs } \binom{8}{k} \text{ times, } k=0,1,\dots,8 & \text{1) } 2 \\ \text{Q) Mean deviation about the mean of the data in P} & \text{2) } \frac{35}{32} \\ \text{R) Variance of the data in which the value } k \text{ occurs } k \text{ times, } k=1,2,\dots,10 & \text{3) } 6 \\ \text{S) Variance of the 15 numbers } a+b \text{, over all pairs } a<b \text{ from } \{1,2,\dots,6\} & \text{4) } \frac{14}{3} \\ & \text{5) } \frac{35}{12} \end{array} Match each entry of List-I with its value in List-II.
Not attempted
A
P-1, Q-2, R-3, S-4
B
P-1, Q-2, R-3, S-5
C
P-3, Q-2, R-1, S-4
D
P-1, Q-5, R-3, S-4
11
For nn paired observations (xi,yi)(x_i, y_i), i=1,…,ni=1,\dots,n, the variance of the numbers xi+yix_i+y_i is 4040, the variance of the numbers xi−yix_i-y_i is 1818, and the variance of the xix_i is 99. Find the standard deviation of the yiy_i, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 4.47
12
Each bar chart shows a data set of 1010 observations taking values in {1,2,3,4,5}\{1,2,3,4,5\} (the number above each bar is its frequency). For which data set is the mean deviation about the median EQUAL to the standard deviation?
Not attempted
A
Values 2 and 4 with frequencies 4 and 6
xfreq1024304650
B
Values 1, 3, 5 with frequencies 3, 4, 3
xfreq1320344053
C
Values 2 and 4, five times each
xfreq1025304550
D
Values 1, 2, 4, 5 with frequencies 1, 4, 4, 1
xfreq1124304451
13
For n≥2n\ge 2 real observations x1,…,xnx_1,\dots,x_n (not all equal) with mean xˉ\bar x, let δ=1n∑∣xi−xˉ∣\delta=\frac1n\sum|x_i-\bar x| (mean deviation about the mean) and σ2=1n∑(xi−xˉ)2\sigma^2=\frac1n\sum(x_i-\bar x)^2. Which statements are true for EVERY such data set?
Not attempted
A
The variance of the nn numbers ∣x1−xˉ∣,…,∣xn−xˉ∣|x_1-\bar x|,\dots,|x_n-\bar x| equals σ2−δ2\sigma^2-\delta^2
B
σ≤2 δ\sigma\le\sqrt2\,\delta
C
The mean deviation about the mean of 2x1+3,…,2xn+32x_1+3,\dots,2x_n+3 equals 2δ+32\delta+3
D
σ≤n2 δ\sigma\le\sqrt{\frac n2}\,\delta
14
For each of the 120120 three-element subsets of {1,2,…,10}\{1,2,\dots,10\} write down its largest element. Find the variance of these 120120 numbers, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 2.89
15
The integers 1,2,…,81,2,\dots,8 are arranged in a row in a uniformly random order. Find the probability that no two consecutive integers (ii and i+1i+1, in either order) occupy adjacent positions. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.13
16
A point is chosen uniformly at random from S={1,2,…,12}S=\{1,2,\dots,12\}. Let A={1,2,3,4,5,6}A=\{1,2,3,4,5,6\} and B={1,2,3,7,8,9}B=\{1,2,3,7,8,9\} (so AA and BB are independent). Find the number of events C⊆SC\subseteq S (including ∅\varnothing and SS) such that CC is independent of AA and CC is independent of BB, but AA, BB, CC are NOT mutually independent.
Not attempted
Your answer: —
Correct answer: 236
17
A bag has four coins: two fair coins, one coin with heads on both faces, and one coin that shows heads with probability 13\frac13. A coin is drawn at random and tossed twice; both tosses show heads. The same coin is tossed a third time. What is the probability that the third toss shows heads?
Not attempted
A
712\frac{7}{12}
B
139174\frac{139}{174}
C
2942\frac{29}{42}
D
139432\frac{139}{432}
18
A coin with P(head)=0.6P(\text{head})=0.6 is tossed repeatedly. Find the probability that the pattern THH (tail, head, head on three consecutive tosses) appears before the pattern HHT.
Not attempted
Your answer: —
Correct answer: 0.64
19
A permutation σ\sigma of {1,2,3,4,5,6}\{1,2,3,4,5,6\} is chosen uniformly at random. Call ii a fixed point if σ(i)=i\sigma(i)=i, call position kk a record if σ(k)\sigma(k) is larger than all of σ(1),…,σ(k−1)\sigma(1),\dots,\sigma(k-1) (position 11 is always a record), and call a pair i<ji<j an inversion if σ(i)>σ(j)\sigma(i)>\sigma(j). Which statements are correct?
Not attempted
A
The variance of the number of fixed points is 11
B
The probability of exactly one fixed point is greater than the probability of no fixed point
C
The expected number of inversions is 152\frac{15}{2}
D
The expected number of records is 4920\frac{49}{20}
20
A biased coin with P(head)=p<12P(\text{head})=p<\frac12 is tossed 66 times. The probability that the number of heads is even (zero counts as even) is 365729\frac{365}{729}. Find the probability of getting exactly 22 heads in these 66 tosses, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.33
21
Two numbers xx and yy are chosen independently and uniformly at random from the interval [0,1][0,1]. Given that segments of lengths xx, yy and 11 can form a (non-degenerate) triangle, find the probability that this triangle is obtuse. Give the answer correct to two decimal places (take π=3.1416\pi=3.1416).
Not attempted
Your answer: —
Correct answer: 0.57
22
In a town, if it rains on a day, it rains on the next day with probability 0.70.7; if it does not rain on a day, it rains on the next day with probability 0.20.2 (this depends only on the previous day). The probability that it rains on a particular Monday is 0.30.3. Given that it rains on the following Thursday, find the probability that it rained on that Monday, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.37
23
Three fair dice are rolled. You are told only that at least one of the dice shows the number 22. What is the probability that all three dice show even numbers?
Not attempted
A
2791\frac{27}{91}
B
14\frac14
C
18\frac18
D
1991\frac{19}{91}
24
Five married couples (ten people) are seated at random in the ten seats around the circular table shown (seat 1010 is next to seat 11), all circular arrangements being equally likely. Find the probability that no husband sits next to his wife. Give the answer correct to two decimal places.
Not attempted
12345678910table
Your answer: —
Correct answer: 0.31
25
Paragraph I An urn contains 33 red and 22 blue balls. A ball is drawn at random, its colour is noted, and it is returned to the urn TOGETHER WITH 22 more balls of the same colour. This procedure is repeated, so that the number of balls in the urn grows by 22 after every draw. Question 1 of 2: Find the probability that the ball obtained on the fifth draw is red. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.6
26
Paragraph I An urn contains 33 red and 22 blue balls. A ball is drawn at random, its colour is noted, and it is returned to the urn TOGETHER WITH 22 more balls of the same colour. This procedure is repeated, so that the number of balls in the urn grows by 22 after every draw. Question 2 of 2: Given that the ball obtained on the third draw is blue, find the probability that the ball obtained on the first draw was red. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.43
27
Paragraph II Players A and B play a sequence of independent rounds. In each round A wins with probability 23\frac23 and B wins with probability 13\frac13; the loser of a round gives one coin to the winner. A starts with 33 coins and B with 55 coins, and play stops as soon as one player has all 88 coins. Question 1 of 2: Find the probability that A ends up with all 88 coins. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.88
28
Paragraph II Players A and B play a sequence of independent rounds. In each round A wins with probability 23\frac23 and B wins with probability 13\frac13; the loser of a round gives one coin to the winner. A starts with 33 coins and B with 55 coins, and play stops as soon as one player has all 88 coins. Question 2 of 2: Given that A wins the first round, find the probability that A ends up with all 88 coins. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.94
29
Let AA, BB, CC be events in a probability space. Which of the following statements are true?
Not attempted
A
If P(A)=P(B)=P(C)=12P(A)=P(B)=P(C)=\frac12 and A,B,CA,B,C are pairwise independent, then P(A∪B∪C)P(A\cup B\cup C) can be as small as 12\frac12
B
If AA is independent of BB, AA is independent of CC, and BB, CC are mutually exclusive, then AA is independent of B∪CB\cup C
C
If A,B,CA,B,C are pairwise independent and AA is independent of B∩CB\cap C, then A,B,CA,B,C are mutually independent
D
If AA is independent of BB and AA is independent of CC, then AA is independent of B∩CB\cap C
30
A fair die is rolled 66 times. Find the expected number of different faces that appear (for example, the outcome 3,3,5,1,5,33,3,5,1,5,3 shows 33 different faces).
Not attempted
A
6(56)66\left(\frac56\right)^6
B
6(1−(56)6)6\left(1-\left(\frac56\right)^6\right)
C
1+(56)+⋯+(56)51+\left(\frac56\right)+\dots+\left(\frac56\right)^5
D
6−(56)66-\left(\frac56\right)^6
31
A fair die is rolled until a six appears or until 55 rolls have been made, whichever happens first. What is the expected number of rolls made?
Not attempted
A
46511296\frac{4651}{1296}
B
122817776\frac{12281}{7776}
C
122814651\frac{12281}{4651}
D
66
32
A random variable XX has a binomial distribution B(n,p)B(n,p). Its most probable values are exactly 66 and 77 (both equally likely, and more likely than any other value), and its variance is 409\frac{40}{9}. Which statements are correct?
Not attempted
A
P(X≥7)>12P(X\ge 7)>\frac12
B
P(X is even)=12(1+3−20)P(X\text{ is even})=\frac12\left(1+3^{-20}\right)
C
p=720p=\frac{7}{20}
D
n=20n=20
33
A game show has 55 closed doors; 22 hide cars and 33 hide goats. You pick one door. The host, who knows what is behind each door, then opens one of the 44 doors you did not pick, choosing uniformly at random among those that hide goats. You now switch to one of the 33 remaining closed doors (not your original choice and not the opened one), chosen uniformly at random. What is the probability that you win a car?
Not attempted
A
23\frac23
B
25\frac25
C
815\frac{8}{15}
D
12\frac12
34
Three distinct vertices are chosen at random from the 1212 vertices V1,…,V12V_1,\dots,V_{12} of a regular 1212-gon (figure), all (123)\binom{12}{3} choices being equally likely. Find the probability that the triangle formed is acute-angled. Give the answer correct to two decimal places.
Not attempted
V1V2V3V4V5V6V7V8V9V10V11V12
Your answer: —
Correct answer: 0.18
35
A disease affects 1%1\% of a population. A test gives a positive result for 95%95\% of people who have the disease and for 10%10\% of people who do not. When the test is repeated on the same person, the results are independent of each other given the person's true condition. A randomly chosen person is tested repeatedly. Find the least number of consecutive positive results after which the probability that the person has the disease exceeds 0.90.9.
Not attempted
A
22
B
33
C
55
D
44
36
Events AA, BB, CC satisfy P(A)=0.6P(A)=0.6, P(B)=0.7P(B)=0.7, P(C)=0.8P(C)=0.8, and nothing else is known about them. Which statements are correct?
Not attempted
A
P(A∩B∩C)P(A\cap B\cap C) can be as large as 0.60.6
B
The probability that exactly one of A,B,CA,B,C occurs can be 0.50.5
C
P(A∪B∪C)P(A\cup B\cup C) can be as small as 0.80.8
D
P(A∩B∩C)≥0.1P(A\cap B\cap C)\ge 0.1 always, and the value 0.10.1 is possible
37
Four letters are placed at random into their four addressed envelopes, one letter per envelope, all 4!4! placements being equally likely. Let XX be the number of letters that go into their correct envelopes. Which bar chart shows the probability distribution of XX (the number above each bar is P(X=k)P(X=k))?
Not attempted
A
P(X=0)=9/24, P(X=1)=8/24, P(X=2)=6/24, P(X=3)=0, P(X=4)=1/24
kP(X=k)03/811/321/43041/24
B
P(X=0)=8/24, P(X=1)=9/24, P(X=2)=6/24, P(X=3)=0, P(X=4)=1/24
kP(X=k)01/313/821/43041/24
C
P(X=0)=9/24, P(X=1)=8/24, P(X=2)=6/24, P(X=3)=1/24, P(X=4)=0
kP(X=k)03/811/321/431/2440
D
P(X=0)=8/24, P(X=1)=8/24, P(X=2)=6/24, P(X=3)=1/24, P(X=4)=1/24
kP(X=k)01/311/321/431/2441/24
38
Three fair dice are rolled and MM is the median (middle value) of the three numbers shown (for example, the median of 5,2,55,2,5 is 55). Which graph shows the cumulative distribution function F(x)=P(M≤x)F(x)=P(M\le x)? (Filled dots show the value of FF at each jump.)
Not attempted
A
Graph through 0, 1/5, 1/2, 4/5, 1, 1 at x = 1, ..., 6
xF(x)1234561/21
B
Graph through 1/216, 8/216, 27/216, 64/216, 125/216, 1 at x = 1, ..., 6
xF(x)1234561/21
C
Graph through 91/216, 152/216, 189/216, 208/216, 215/216, 1 at x = 1, ..., 6
xF(x)1234561/21
D
Graph through 16/216, 56/216, 108/216, 160/216, 200/216, 1 at x = 1, ..., 6
xF(x)1234561/21
39
Five cards are dealt from a well-shuffled standard deck of 5252 cards. Let uu be the probability that the hand has at least two aces given that it has at least one ace, and let vv be the probability that the hand has at least two aces given that it contains the ace of spades. Which statements are correct?
Not attempted
A
u=vu=v, because knowing which ace is held gives no extra information
B
u<vu<v
C
u>18u>\frac18
D
v=1−(484)(514)v=1-\frac{\binom{48}{4}}{\binom{51}{4}}
40
In an election, candidate A receives 1212 votes and candidate B receives 77 votes. The 1919 ballots are counted one at a time in a uniformly random order. Find the probability that at every stage of the counting A's running total is at least as large as B's (ties are allowed). Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.46
41
An urn contains 55 balls, each red or white. The number of red balls is unknown, and each of the values 0,1,2,3,4,50,1,2,3,4,5 is equally likely a priori. Two balls are drawn WITH replacement and both are red. A third ball is then drawn (again with replacement). What is the probability that it is red?
Not attempted
A
34\frac34
B
911\frac{9}{11}
C
35\frac{3}{5}
D
1115\frac{11}{15}
42
Six red and five blue balls are arranged in a row in a uniformly random order (balls of the same colour are identical). A run is a maximal block of consecutive balls of the same colour (for example RRBBRBBRRRB has 66 runs). Find the variance of the number of runs, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 2.43
43
Three fair dice are rolled. List-IList-IIP) Probability that the product of the three numbers is a perfect square1) 19108Q) Probability that the sum of the three numbers is 102) 18R) Probability that the three numbers can be arranged as an A.P. (a constant triple counts)3) 736S) Probability that the largest minus the smallest number is exactly 24) 295) 16\begin{array}{ll}\textbf{List-I} & \textbf{List-II} \\ \text{P) Probability that the product of the three numbers is a perfect square} & \text{1) } \frac{19}{108} \\ \text{Q) Probability that the sum of the three numbers is } 10 & \text{2) } \frac{1}{8} \\ \text{R) Probability that the three numbers can be arranged as an A.P. (a constant triple counts)} & \text{3) } \frac{7}{36} \\ \text{S) Probability that the largest minus the smallest number is exactly } 2 & \text{4) } \frac{2}{9} \\ & \text{5) } \frac{1}{6} \end{array} Match each entry of List-I with its value in List-II.
Not attempted
A
P-3, Q-2, R-1, S-4
B
P-1, Q-5, R-3, S-4
C
P-1, Q-2, R-3, S-4
D
P-1, Q-2, R-4, S-3
44
Players A and B toss fair coins independently and count their heads. Which statements are correct?
Not attempted
A
If each tosses 55 times, the probability that A gets strictly more heads than B is 193512\frac{193}{512}
B
If each tosses 55 times, the probability that they get the same number of heads equals the probability of exactly 55 heads in 1010 tosses of one fair coin
C
If A tosses 66 times and B tosses 55 times, the probability that A gets strictly more heads than B is 12\frac12
D
If A tosses 77 times and B tosses 55 times, the probability that A gets strictly more heads than B is 34\frac34
45
Three marksmen A, B, C fire at a target in the repeating order A, B, C, A, B, C, ... Each shot of A, B, C hits with probability 12\frac12, 13\frac13, 14\frac14 respectively, independently of all other shots. Firing stops as soon as the target has been hit twice in total (by anyone). Find the probability that the shot which makes the second hit is fired by B. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.36
46
Three subsets AA, BB, CC of S={1,2,3,4,5,6}S=\{1,2,3,4,5,6\} are chosen independently, each uniformly at random from all 6464 subsets of SS. What is the probability that A∩B∩C=∅A\cap B\cap C=\varnothing and A∪B∪C=SA\cup B\cup C=S?
Not attempted
A
(34)6\left(\frac34\right)^6
B
(78)12\left(\frac78\right)^{12}
C
(78)6⋅12\left(\frac78\right)^6\cdot\frac{1}{2}
D
(78)6−(18)6\left(\frac{7}{8}\right)^{6}-\left(\frac18\right)^{6}
47
Six students put their own pens in a box and then each draws one pen at random (all 6!6! assignments equally likely). Find the probability that no student gets his own pen AND no two students get each other's pens. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.22
48
An urn contains 55 red and 33 black balls. Balls are drawn one at a time at random without replacement, and drawing stops as soon as all 33 black balls have been drawn. Find the expected number of balls drawn.
Not attempted
Your answer: —
Correct answer: 6.75
49
Two hospitals treat a disease (the bar chart shows the recovery rates). Hospital P treated 100100 mild cases (9090 recovered) and 100100 severe cases (3030 recovered). Hospital Q treated 200200 mild cases (170170 recovered) and 2020 severe cases (55 recovered). One of the 420420 patients is chosen at random. Which statements are correct?
Not attempted
recovery rate190/100P170/200Qmild cases30/100P5/20Qsevere cases
A
The probability of recovery given 'treated at P and mild' exceeds that given 'treated at Q and mild', and the same comparison holds for severe cases
B
Given that the chosen patient recovered, the probability that the patient was treated at P is 2459\frac{24}{59}
C
Had Q's severe cases recovered at P's severe-case rate, Q's overall recovery probability would still be below P's
D
The probability of recovery given 'treated at P' exceeds the probability of recovery given 'treated at Q'
50
A fair die is rolled 1111 times. Given that the face 66 appeared exactly 33 times, find the probability that no two of the sixes occurred on consecutive rolls. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.51
51
For events AA, BB, CC: P(A∪B)=0.60P(A\cup B)=0.60, P(B∪C)=0.65P(B\cup C)=0.65, P(C∪A)=0.70P(C\cup A)=0.70, P(A∪B∪C)=0.80P(A\cup B\cup C)=0.80 and P(A∩B∩C)=0.05P(A\cap B\cap C)=0.05. Find the probability of the shaded event in the figure, i.e. that exactly two of AA, BB, CC occur.
Not attempted
ABC
A
0.350.35
B
0.300.30
C
0.250.25
D
0.400.40
52
Events AA, BB, CC are mutually independent with P(A)=12P(A)=\frac12, P(B)=13P(B)=\frac13, P(C)=14P(C)=\frac14. For which of the shaded events (shown in the four Venn diagrams) is the probability exactly 14\frac14?
Not attempted
A
Exactly one of B, C occurs
ABC
B
Exactly one of A, B occurs and C does not
ABC
C
A occurs together with at least one of B, C
ABC
D
Exactly two of A, B, C occur
ABC
53
XX has the binomial distribution B(5,13)B\left(5,\frac13\right). Which statements are correct?
Not attempted
A
E(2X)=(43)5E\left(2^X\right)=\left(\frac43\right)^5
B
E(1X+1)=1E(X)+1=38E\left(\frac{1}{X+1}\right)=\frac{1}{E(X)+1}=\frac38
C
E(1X+1)=6651458E\left(\frac{1}{X+1}\right)=\frac{665}{1458}
D
E(X(X−1))=409E\left(X(X-1)\right)=\frac{40}{9}
54
Three fair dice are rolled. What is the expected value of (largest number shown) minus (smallest number shown)?
Not attempted
A
52\frac{5}{2}
B
3518\frac{35}{18}
C
3512\frac{35}{12}
D
9136\frac{91}{36}
55
A fair coin is tossed until the pattern H, T, H appears on three consecutive tosses. What is the expected number of tosses?
Not attempted
A
1414
B
88
C
66
D
1010
56
A bag has 1010 balls of 1010 different colours, one of them red. A ball is drawn at random. Two witnesses A and B independently see the ball. A reports its colour truthfully with probability 34\frac34, B with probability 45\frac45; a witness who lies names one of the other 99 colours chosen at random. Both report 'red'. What is the probability that the ball is red?
Not attempted
A
108109\frac{108}{109}
B
47\frac47
C
35\frac{3}{5}
D
1213\frac{12}{13}
57
Ten independent trials are performed; the kk-th trial succeeds with probability 12k+1\frac{1}{2k+1} (k=1,2,…,10)(k=1,2,\dots,10). What is the probability that the total number of successes is odd?
Not attempted
A
121\frac{1}{21}
B
12\frac12
C
1121\frac{11}{21}
D
1021\frac{10}{21}
58
The histogram shows the marks of a group of students in the classes 00-1010, 1010-2020, 2020-4040, 4040-6060. The vertical axis is FREQUENCY DENSITY (frequency per mark), with heights 11, 33, 1.51.5, 0.50.5 respectively. Find the mean of the distribution minus its median (using class marks for the mean).
Not attempted
xfreq. density0102040600.511.53
A
00
B
154\frac{15}{4}
C
−54-\frac{5}{4}
D
52\frac{5}{2}
59
Machines M1, M2, M3 produce 50%50\%, 30%30\%, 20%20\% of a factory's bolts; M1 and M2 produce defective bolts at rates 2%2\% and 3%3\%, and M3's rate (marked ? in the tree) is unknown. Overall, 3%3\% of the bolts are defective. All bolts in a box come from one machine, the box's machine being M1, M2, M3 with the probabilities above, and bolts are defective independently given the machine. A bolt drawn from a box is defective. What is the probability that a second bolt from the same box is also defective?
Not attempted
0.50.30.2M1M2M30.020.03?defectivegooddefectivegooddefectivegood
A
1130\frac{11}{30}
B
3100\frac{3}{100}
C
431200\frac{43}{1200}
D
11200\frac{11}{200}
60
A frog sits at vertex A of the regular hexagon ABCDEF shown. Every second it jumps to one of the two vertices adjacent to its current vertex, each with probability 12\frac12, independently of earlier jumps. What is the expected number of jumps until it first reaches the opposite vertex D?
Not attempted
ABCDEF
A
33
B
66
C
99
D
1212
61
In the network shown, each of the five switches (numbered circles: 11 on S-X, 22 on S-Y, 33 on X-T, 44 on Y-T, 55 on X-Y) is closed with probability 23\frac23, independently of the others. Current can flow along a link only through a closed switch, in either direction. What is the probability that current can flow from S to T?
Not attempted
12345STXY
A
5002459049\frac{50024}{59049}
B
5681\frac{56}{81}
C
6481\frac{64}{81}
D
184243\frac{184}{243}
62
On the 4×44\times4 grid shown, a path from O(0,0)O(0,0) to (4,4)(4,4) uses unit steps right or up only; one of the (84)=70\binom84=70 such paths is chosen uniformly at random. Which statements are correct?
Not attempted
O(0,0)(4,4)(2,2)
A
The expected number of diagonal points (1,1)(1,1), (2,2)(2,2), (3,3)(3,3) on the path is 5835\frac{58}{35}
B
The probability that the path never goes above the dashed diagonal y=xy=x is 15\frac15
C
If instead each step is chosen right or up by a fair coin (with the remaining steps forced once x=4x=4 or y=4y=4 is reached), the probability of passing through (2,2)(2,2) is again 1835\frac{18}{35}
D
The probability that the path passes through (2,2)(2,2) is 1835\frac{18}{35}
63
A random variable XX has the distribution shown: P(X=0)=0.1P(X=0)=0.1, P(X=1)=0.2P(X=1)=0.2, P(X=2)=0.4P(X=2)=0.4, P(X=3)=0.2P(X=3)=0.2, P(X=4)=0.1P(X=4)=0.1. Let X1X_1, X2X_2 be independent, each with this distribution. Which statements are correct?
Not attempted
kP(X=k)012340.10.20.4
A
E[(X1−X2)2]=2.4E\left[(X_1-X_2)^2\right]=2.4
B
P(X1=X2)=0.26P(X_1=X_2)=0.26
C
P(X1>X2)=0.37P(X_1>X_2)=0.37
D
E∣X1−X2∣=1.2E|X_1-X_2|=1.2
64
The bar charts show two groups of 2020 observations each: Group I takes values 1,2,3,4,51,2,3,4,5 with frequencies 2,4,8,4,22,4,8,4,2 and Group II takes values 4,5,6,7,84,5,6,7,8 with frequencies 2,4,8,4,22,4,8,4,2. The two groups are combined into one data set of 4040 observations. Which statements are correct?
Not attempted
xf1224384452Group Ixf4254687482Group II
A
The variance of the combined data equals the average of the two group variances
B
The combined data have a unique mode
C
The variance of the combined data is 3.453.45
D
The coefficient of variation of Group I is twice that of Group II
65
A function f:{1,2,3,4}→{1,2,3,4}f:\{1,2,3,4\}\to\{1,2,3,4\} is chosen uniformly at random from all 44=2564^4=256 such functions. Which statements are correct?
Not attempted
A
P(f∘f=identity)=512P(f\circ f=\text{identity})=\frac{5}{12}
B
P(the range of f has exactly 2 elements)=2164P(\text{the range of }f\text{ has exactly }2\text{ elements})=\frac{21}{64}
C
P(f has no fixed point)=81256P(f\text{ has no fixed point})=\frac{81}{256}
D
P(f∘f=f)=41256P(f\circ f=f)=\frac{41}{256}
66
For events with all the conditional probabilities below defined, which statements are ALWAYS true?
Not attempted
A
P(A∣B)+P(A∣B′)=1P(A\mid B)+P(A\mid B')=1
B
If P(A∣C)>P(B∣C)P(A\mid C)>P(B\mid C) and P(A∣C′)>P(B∣C′)P(A\mid C')>P(B\mid C'), then P(A)>P(B)P(A)>P(B)
C
If P(A∣B)>P(A)P(A\mid B)>P(A) and P(B∣C)>P(B)P(B\mid C)>P(B), then P(A∣C)>P(A)P(A\mid C)>P(A)
D
If P(A∣B)>P(A)P(A\mid B)>P(A), then P(B′∣A)>P(B′)P(B'\mid A)>P(B')
67
List-I (variance of)List-IIP) the number of fixed points of a random permutation of {1,…,7}1) 1Q) the number of sixes in 4 rolls of a fair die2) 59R) the number of red balls when 4 balls are drawn without replacement from 5 red and 5 white3) 23S) the sum of the numbers on two fair dice4) 3565) 3512\begin{array}{ll}\textbf{List-I (variance of)} & \textbf{List-II} \\ \text{P) the number of fixed points of a random permutation of } \{1,\dots,7\} & \text{1) } 1 \\ \text{Q) the number of sixes in } 4 \text{ rolls of a fair die} & \text{2) } \frac{5}{9} \\ \text{R) the number of red balls when } 4 \text{ balls are drawn without replacement from } 5 \text{ red and } 5 \text{ white} & \text{3) } \frac{2}{3} \\ \text{S) the sum of the numbers on two fair dice} & \text{4) } \frac{35}{6} \\ & \text{5) } \frac{35}{12} \end{array} Match each random variable in List-I with its variance in List-II.
Not attempted
A
P-1, Q-2, R-1, S-4
B
P-1, Q-2, R-3, S-4
C
P-3, Q-2, R-1, S-5
D
P-1, Q-3, R-2, S-4
68
List-I (expected value of)List-IIP) the number of empty boxes when 5 distinct balls are put at random into 5 boxes1) 1024625Q) the number of i∈{2,…,6} with σ(i)>σ(i−1), σ(i)>σ(i+1), for a random permutation σ of {1,…,7}2) 53R) the number of couples sitting together when 4 couples sit at random around a round table3) 87S) the number of different values shown when 3 fair dice are rolled4) 91365) 43\begin{array}{ll}\textbf{List-I (expected value of)} & \textbf{List-II} \\ \text{P) the number of empty boxes when } 5 \text{ distinct balls are put at random into } 5 \text{ boxes} & \text{1) } \frac{1024}{625} \\ \text{Q) the number of } i\in\{2,\dots,6\} \text{ with } \sigma(i)>\sigma(i-1),\ \sigma(i)>\sigma(i+1) \text{, for a random permutation } \sigma \text{ of } \{1,\dots,7\} & \text{2) } \frac{5}{3} \\ \text{R) the number of couples sitting together when } 4 \text{ couples sit at random around a round table} & \text{3) } \frac{8}{7} \\ \text{S) the number of different values shown when } 3 \text{ fair dice are rolled} & \text{4) } \frac{91}{36} \\ & \text{5) } \frac{4}{3} \end{array} Match each entry of List-I with its value in List-II.
Not attempted
A
P-1, Q-2, R-3, S-4
B
P-1, Q-5, R-3, S-4
C
P-4, Q-2, R-3, S-1
D
P-1, Q-2, R-5, S-4
69
A family has two children. Each child is independently a boy or a girl with probability 12\frac12, is born on each of the 77 days of the week with probability 17\frac17, and in each of the 1212 months with probability 112\frac1{12}. Find the probability that both children are boys given the information in List-I. List-I (information)List-IIP) at least one child is a boy1) 13Q) the elder child is a boy2) 12R) at least one child is a boy born on a Sunday3) 1327S) at least one child is a boy born in January or February4) 11235) 14\begin{array}{ll}\textbf{List-I (information)} & \textbf{List-II} \\ \text{P) at least one child is a boy} & \text{1) } \frac{1}{3} \\ \text{Q) the elder child is a boy} & \text{2) } \frac{1}{2} \\ \text{R) at least one child is a boy born on a Sunday} & \text{3) } \frac{13}{27} \\ \text{S) at least one child is a boy born in January or February} & \text{4) } \frac{11}{23} \\ & \text{5) } \frac{1}{4} \end{array}
Not attempted
A
P-5, Q-2, R-3, S-4
B
P-1, Q-2, R-2, S-4
C
P-1, Q-2, R-3, S-4
D
P-1, Q-2, R-3, S-2
70
The data 102,104,107,109,113102, 104, 107, 109, 113 have coefficient of variation c1=σxˉc_1=\frac{\sigma}{\bar x}. Each observation xx is replaced by 3x−3003x-300, and the new data have coefficient of variation c2c_2. What is c2c1\frac{c_2}{c_1}?
Not attempted
A
3217\frac{321}{7}
B
33
C
11
D
1077\frac{107}{7}
Scaling multiplies both σ\sigma and xˉ\bar x, but a change of origin moves only the mean: the coefficient of variation is NOT invariant under y=ax+by=ax+b with b≠0b\ne0.
Mean xˉ=107\bar x=107. Under y=3x−300y=3x-300: yˉ=3⋅107−300=21\bar y=3\cdot107-300=21 and σy=3σx\sigma_y=3\sigma_x. c1=σx107c_1=\frac{\sigma_x}{107}, c2=3σx21=σx7c_2=\frac{3\sigma_x}{21}=\frac{\sigma_x}{7}. c2c1=1077≈15.3\frac{c_2}{c_1}=\frac{107}{7}\approx15.3; the actual value of σx\sigma_x (14.8\sqrt{14.8}) is not needed. Traps: 'σxˉ\frac{\sigma}{\bar x} scales like σ\sigma' gives 33; 'CV is unit-free, so unchanged' gives 11 (true only for b=0b=0); 3217\frac{321}{7} multiplies the correct ratio by 33 again. Answer: 1077\frac{107}{7}.